There were commercial computers that did. The PDP-8, as one example (not 74xx, but similar).
Edit: The Xerox Alto CPU was built with 74181 chips.
Cost, power consumption, space, etc, are all disadvantages. I assume the Gigatron is using modern 74HCxx CMOS chips, which consume less power than the original 74xx TTL chips.
Edit: The Xerox Alto CPU was built with 74181 chips.
Cost, power consumption, space, etc, are all disadvantages. I assume the Gigatron is using modern 74HCxx CMOS chips, which consume less power than the original 74xx TTL chips.